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Lab 2 slides - Review 1 N(1 2/SXX 1 0 N(0 2 n SSreg/1...

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Review ˆ β 1 N ( β 1 , σ 2 /S XX ) ˆ β 0 N ( β 0 , σ 2 ( 1 n + ¯ x 2 S XX )) SS reg / 1 SS res / ( n - 2) ∼ F 1 ,n - 2 Construction of t - intervals for β 1 and β 0 individually Lecture 4 – p. 1/2
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Assessing Fit Will later examine many different criteria for model fit and model selection Most basic: how much of the variability of the response is due to the variability in the covariate (or explanatory or independent) variable Natural measure is the coefficient of determination (or R 2 ): R 2 = SS Reg SS Tot Could also write as: R 2 = 1 - SS Res SS Tot Lecture 4 – p. 2/2
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Properties of R 2 Obviously between 0 and 1 Larger values indicate more explanatory power of the covariate Does not reflect appropriateness of the model assumptions or the causal relationship of the covariate to the response (except in rare cases) Interpretation of R 2 varies depending on the scientific environment and application and goals of the analysis R 2 can be made artificially high by increasing S XX (and therefore the contribution of SS Reg to SS Tot ) with a few carefully chosen points (remember that E ( SS Reg ) = σ 2 + β 2 1 S XX Lecture 4 – p. 3/2
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Accounting data > summary(account.mod) Call: lm(formula = MARKETRATE ˜ ACCOUNTINGRATE) Residuals: Min 1Q Median 3Q Max -6.396 -3.307 -1.294 2.837 13.487 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 0.8480 1.9765 0.429 0.67 ACCOUNTINGRATE 0.6103 0.1432 4.263 8.5e-05 *** --- Signif. codes: 0 ’ *** ’ 0.001 ’ ** ’ 0.01 ’ * ’ 0.05 ’.’ 0.1 ’ ’ 1 Residual standard error: 5.086 on 52 degrees of freedom Multiple R-Squared: 0.259, Adjusted R-squared: 0.2448 F-statistic: 18.18 on 1 and 52 DF, p-value: 8.505e-05 Lecture 4 – p. 4/2
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Two kinds of prediction Extremely important to distinguish between two types of predicted values Predicted mean of the response for a particular x 0 value Predicted value of the response for a particular x 0 value Although the numeric predictions are the same, the nature of uncertainty is very different Lecture 4 – p. 5/2
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Predicted mean response For any ˆ β 0 and ˆ β 1 and x 0 value we have ˆ y ( x 0 ) = ˆ β 0 + ˆ β 1 x 0 E y ( x 0 )) is obviously β 0 + β 1 x 0 by linearity of expectation Can show V ar y ( x 0 )) = V ar y + b 1 ( x 0 - ¯ x )) = σ 2 ( 1 n + ( x 0 - ¯ x ) 2 S XX ) NOTE: This gives point-wise confidence bands for the mean of the response, not a confidence band for the actual respone values!
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